Parametric representations and quasiconformal extensions by means - - PowerPoint PPT Presentation

Spaces of Analytic Functions and Singular Integrals 2016

Parametric representations and quasiconformal extensions by means of modern Loewner Theory Pavel Gumenyuk

University of Stavanger, Norway

• St. Petersburg – RUSSIA, October 17–20, 2016

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Introduction

Classical Parametric Representation

Every univalent holomorphic function f : D :=

• z : |z| < 1
• → C,

f(0) = 0, f′(0) = 1, is the initial element, i.e. f = f0,

• f some (classical radial) Loewner chain (ft)t0.

Definition: (ft)t0 is a classical radial Loewner chain if:

(i) for each t 0, ft : D → C is univalent in D, ft(0) = 0, f′

t (0) = et;

(ii) fs(D) ⊂ ft(D) whenever 0 s t.

Loewner – Kufarev PDE

∂ft ∂t = −G(z, t)∂ft ∂z , t 0, G(z, t) := −z p(z, t), where p is a classical Herglotz function.

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Introduction

Loewner – Kufarev PDE

∂ft ∂t = −G(z, t) ∂ft ∂z , G(z, t) := −z p(z, t), (L-K PDE) where p is a classical Herglotz function, i.e. (i) ∀z ∈ D, p(z, t) is measurable in t ∈ [0, +∞); (ii) ∀ a.e. t 0, p(·, t) is holomorphic in D, Re p > 0, p(0, t) = 1. (L-K PDE) establish a 1-to-1 relation between

• class. Herglotz functions p

and

• class. radial Loewner chains (ft).

Loewner – Kufarev ODE = characteristic eq-n for (L-K PDE)

d dt ϕs,t(z) = G

• ϕs,t(z), t
• ,

t s 0; ϕs,s(z) = z ∈ D. ϕs,t = f−1

t

• fs : D

holo

−→ D

because

fs(D) ⊂ ft(D), t s 0.

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Q.c.-extensions via classical Loewner chains

General question

How are the properties of a classical Herglotz function p reflected in properties of the corresponding Loewner chain (ft) and evolution family (ϕs,t)ts0?

Theorem (J. Becker, 1972)

Let k ∈ [0, 1). SUPPOSE that the class. Herglotz function p satisfies p(D, t) ⊂ U(k) :=

• ζ:
• ζ − 1

ζ + 1

• k
• ⊂⊂ H

a.e. t 0.

THEN each function in the corresponding Loewner chain (ft) have a k-q.c. extension to C. NB: Becker also gave an explicit formula for the q.c.-extension of ft’s.

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Generalized Loewner – Kufarev ODE

F . Bracci, M.D. Contreras, and S. Díaz-Madrigal, 2008/2012

d dt ϕs,t(z) = G

• ϕs,t(z), t
• ,

t s 0; ϕs,s(z) = z ∈ D, (∗) where G(w, t) :=

• τ(t) − w
• 1 − τ(t)w
• p(w, t)

and: (i) τ : [0, +∞) → D is measurable; (ii) ∀z ∈ D, p(w, t) is measurable in t ∈ [0, +∞); (iii) ∀ a.e. t 0, p(·, t) is holomorphic in D with Re p 0; (iv) t → p(0, t) is L1

loc on [0, +∞).

The function G above is referred to as a Herglotz vector field and (ϕs,t)ts0 is the associated evolution family.

Classical case:

τ(t) = 0 and p(0, t) = 1 for a.e. t 0.

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Generalized Loewner Chains

Definition: (ft) is a (generalized) Loewner chain if:

(i) for each t 0, ft : D → C is univalent in D; (ii) fs(D) ⊂ ft(D) whenever 0 s t; (iii)∀ K ⊂⊂ D, supz∈K |ft(z) − fs(z)| t

s

αK(ξ) dξ for any s t and some L1

loc function αK : [0, +∞) → [0, +∞).

Theorem (M.D. Contreras, S. Díaz-Madrigal, and P

. Gum., 2010)

Let G be a Herglotz vector field with associated evol’n family (ϕs,t). There exists a unique Loewner chain (ft) such that (i) ϕs,t = f−1

t

• fs whenever t s 0; (ii)

f0(0) = 0, f′

0(0) = 1;

(iii)

• t0 ft(D) is C or a disk centered at 0.

∂ft/∂t = −G(z, t) ∂ft/∂z , t 0. Moreover, (gL-K PDE)

We call (ft) the standard Loewner chain associated with (ϕs,t) and G.

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Q.c.-extendibility of gen’d Loewner chains

Corollary

Let (ϕs,t) be an evol’n family with associated st. Loewner chain (ft). If ϕs,t is k-q.c. extendible for any t s 0, then ft is also k-q.c. extendible for any t 0.

P . Gum., I. Prause, 2016

Becker’s condition p(D, t) ⊂ U(k) is also sufficient for k-q.c. extendibility of evolution families in the general case.

Notation

Let a : [0, +∞) → H ∪ ˙ ιR be L1

loc and denote by Dt: ◮ the (closed) hyperbolic disk in H of radius 1

2 log 1+k 1−k centered at a(t)

when a(t) ∈ H;

◮ the single point {a(t)} when a(t) ∈ ˙

ιR.

Remark: If a ≡ 1, then Dt ≡ U(k).

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Q.c.-extendibility of gen’d Loewner chains

Theorem (P . Gum., I. Prause, 2016)

If G(w, t) :=

• τ(t) − w
• 1 − τ(t)w
• p(w, t) is a Herglotz vector field

p(D, t) ⊂ Dt

for a.e. t 0,

and then each ϕs,t in the assoc’d evolution family is k-q.c. extendible. [No explicit formula for the q.c.-extensions; we use Slodkowski’s λ-Lemma] SPECIAL CASE τ ≡ 1 (aka “chordal” case): P . Gum., I. Hotta, 2016 [with explicit formula for the extension]

Theorem (P . Gum., I. Hotta, 2016)

SUPPOSE that h is holomorphic and locally univalent in H, and α h(z)

h′(z) − z

• + iβ

∈ U(k) =

• ζ: |ζ − 1| k|ζ + 1|
• ∀ z ∈ H

(∗), where α > 0 and β ∈ R are some constants, THEN h has a k-q.c. extension to C with a fixed point at ∞.

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Parametric represent’n of univalent self-maps

• Loewner chains provide characterization of

univalent normalized maps f : D → C;

• Similarly, evolution families provide characterization of

univalent self-maps ϕ : D → D: ϕ ∈ U :=

• ϕ ∈ Hol(D, D): ϕ is univalent in D
• if and only if

ϕ belongs to some evolution family (ϕs,t).

Question

Given a subclass U′ ⊂ U, is it possible to characterize ϕ ∈ U′ in a similar way?

Example: classical case

ϕ ∈ U0 :=

• ϕ ∈ U : ϕ(0) = 0, ϕ′(0) > 0
• iff ϕ belongs to some evolution

family (ϕs,t) ∼ a class. Herglotz v.f. G(z, t) = −z p(z, t), Im p(0, t) = 0.

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Parametric represent’n of univalent self-maps

Definition

We say that U′ ⊂ U admits a Loewner-type param. representation, if there exists a convex cone M ⊂

• all Herglotz vector fields
• such that

ϕ ∈ U′ ⇐⇒ ϕ ∈ (ϕs,t) ∼ to some G ∈ M. NB: U′ must be closed w.r.t. (ϕ, ψ) → ϕ ◦ ψ and idD ∈ U′.

Denjoy – Wolff point

Let ϕ ∈ Hol(D, D) \ {idD}. Then:

either

∃! τ ∈ D ϕ(τ) = τ, τ is called the D.W.-point of ϕ

• r ϕ◦n → τ ∈ ∂D, ϕ(τ) := ∠ lim

z→τ ϕ(z) = τ, ϕ′(τ) := ∠ lim z→τ ϕ′(z) ∈ (0, 1].

Boundary regular fixed points

σ ∈ ∂D is a BRFP of ϕ

def

⇐⇒

ϕ (σ) = ∠ limz→σ ϕ(z) = σ and ϕ′(σ) = ∠ limz→σ ϕ′(z) exists finitely.

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Parametric represent’n of univalent self-maps

Let F = {σ1, σ2, . . . , σn} be a finite subset of ∂D, and let τ ∈ D \ F.

• U[F] :=
• ϕ ∈ U : each σ ∈ F is a BRFP of ϕ
• Uτ[F] :=
• ϕ ∈ U[F] \ {idD}: τ is the DW-point of ϕ
• ∪ {idD}

Theorem (P . Gum., arXiv:1603.04043)

The following classes admit a Loewner-type parametric represent’n:

✔ U[F] for n 3; ✔ Uτ[F] for τ ∈ ∂D and n 2; ✔ Uτ[F] for τ ∈ D and any n 1. [n = 1: Unkelbach and Goryainov]

The corresponding cones M of Herglotz vector fields are described.

• H. Unkelbach, 1940: an attempt to give

the Loewner-type parametric representation for U0[{1}]; V.V. Goryainov, 2015: the complete proof.

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